Is "A Short Elementary Proof of the Unprovability of the Collatz Conjecture" really as claimed? The paper "A Short Elementary Proof of the Unprovability of the Collatz Conjecture", from a self-proclaimed peer reviewed journal (although the publisher, ScienceDomain International, appears on Beall's list), purports to prove that the Collatz Conjecture is unprovable.
If it's valid, why has it not received more attention? If it’s invalid, what is the flaw, and why has it not been pointed out since initial publication?
 A: The paper can be quickly identified as flawed by the simple fact that it never discusses which axioms it claims that the Collatz Conjecture cannot be proved from. This alone is proof positive that it cannot do what it purports to do.
If there had been a theorem stating, "The Collatz Conjecture cannot be proved in Peano Arithmetic" or "... in ZFC" or whatever, then some further investigation would be necessary. But the naked assertion that it "cannot be proved", without further specification, is not a meaningful claim. It's not even wrong.
The fact that the flaw is so fundamental also goes some way to explain why nobody has found it worth the effort to point it out in the pages of the journal.
A: I endorse Henning Makholm's points about the lack of any consideration of mathematical logic in the paper.
The form of the argument is also seriously flawed.  The author claims to prove (Theorem 3.3) that the Collatz process will take at least $d$ steps to terminate on input $2^{d+1}-1$. (It's not expressed like that, but that's what it says.) Given this, the author lets $d$ tend to $\infty$ and concludes (Corollary 3.4) that the process will never terminate if the binary expansion of its input comprises an unbounded list of $1$s. (Nowhere is it even mentioned how the Collatz process would proceed on such an input.) That the Collatz conjecture is unprovable is deduced from this "corollary" almost as an aside in the concluding paragraph by conflating a number of the form $n=2^u-1$ with one having "an unbounded number of binary digits".
